Question
If |z|=1 and ω=z−1z+1 (where, z≠−1), then Re (ω) is
Answer: Option A
:
A
Since,|z|=1andω=z−1z+1⇒z−1=ωz+ω⇒z=1+ω1−ω⇒|z|=|1+ω||1−ω|⇒|1−ω|=|1+ω|[∴|z|=1]Onsquaringbothsides,weget1+|ω|2−2Re(ω)=1+|ω|2+2Re(ω)[using|z1±z2|2=|z1|2+|z2|2±2Re(¯z1z2)]⇒4Re(ω)=0⇒Re(ω)=0
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:
A
Since,|z|=1andω=z−1z+1⇒z−1=ωz+ω⇒z=1+ω1−ω⇒|z|=|1+ω||1−ω|⇒|1−ω|=|1+ω|[∴|z|=1]Onsquaringbothsides,weget1+|ω|2−2Re(ω)=1+|ω|2+2Re(ω)[using|z1±z2|2=|z1|2+|z2|2±2Re(¯z1z2)]⇒4Re(ω)=0⇒Re(ω)=0
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